CFRG S. Goldberg
Internet-Draft L. Reyzin
Intended status: Standards Track Boston University
Expires: September 6, 2018 D. Papadopoulos
Hong Kong University of Science and Techology
J. Vcelak
NS1
March 5, 2018
Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-01
Abstract
A Verifiable Random Function (VRF) is the public-key version of a
keyed cryptographic hash. Only the holder of the private key can
compute the hash, but anyone with public key can verify the
correctness of the hash. VRFs are useful for preventing enumeration
of hash-based data structures. This document specifies several VRF
constructions that are secure in the cryptographic random oracle
model. One VRF uses RSA and the other VRF uses Eliptic Curves (EC).
Status of This Memo
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Copyright Notice
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document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3
2. VRF Algorithms . . . . . . . . . . . . . . . . . . . . . . . 4
3. VRF Security Properties . . . . . . . . . . . . . . . . . . . 4
3.1. Full Uniqueness or Trusted Uniqueness . . . . . . . . . . 4
3.2. Full Collison Resistance or Trusted Collision Resistance 5
3.3. Full Pseudorandomness or Selective Pseudorandomness . . . 5
3.4. An additional pseudorandomness property . . . . . . . . . 6
4. RSA Full Domain Hash VRF (RSA-FDH-VRF) . . . . . . . . . . . 7
4.1. RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . . 8
4.2. RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . . 8
4.3. RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . . 9
5. Elliptic Curve VRF (EC-VRF) . . . . . . . . . . . . . . . . . 9
5.1. EC-VRF Proving . . . . . . . . . . . . . . . . . . . . . 11
5.2. EC-VRF Proof To Hash . . . . . . . . . . . . . . . . . . 11
5.3. EC-VRF Verifying . . . . . . . . . . . . . . . . . . . . 12
5.4. EC-VRF Auxiliary Functions . . . . . . . . . . . . . . . 13
5.4.1. EC-VRF Hash To Curve . . . . . . . . . . . . . . . . 13
5.4.2. EC-VRF Hash Points . . . . . . . . . . . . . . . . . 14
5.4.3. EC-VRF Decode Proof . . . . . . . . . . . . . . . . . 15
5.5. EC-VRF Ciphersuites . . . . . . . . . . . . . . . . . . . 15
5.6. When the EC-VRF Keys are Untrusted . . . . . . . . . . . 17
5.6.1. EC-VRF Validate Key . . . . . . . . . . . . . . . . . 17
6. Implementation Status . . . . . . . . . . . . . . . . . . . . 18
7. Security Considerations . . . . . . . . . . . . . . . . . . . 18
7.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 18
7.1.1. Uniqueness and collision resistance with untrusted
keys . . . . . . . . . . . . . . . . . . . . . . . . 18
7.1.2. Pseudorandomness with untrusted keys . . . . . . . . 19
7.2. Selective vs Full Pseudorandomness . . . . . . . . . . . 19
7.3. Proper randomness for EC-VRF . . . . . . . . . . . . . . 20
7.4. Timing attacks . . . . . . . . . . . . . . . . . . . . . 20
8. Change Log . . . . . . . . . . . . . . . . . . . . . . . . . 20
9. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 20
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.1. Normative References . . . . . . . . . . . . . . . . . . 21
10.2. Informative References . . . . . . . . . . . . . . . . . 21
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Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction
1.1. Rationale
A Verifiable Random Function (VRF) [MRV99] is the public-key version
of a keyed cryptographic hash. Only the holder of the private VRF
key can compute the hash, but anyone with corresponding public key
can verify the correctness of the hash.
A key application of the VRF is to provide privacy against offline
enumeration (e.g. dictionary attacks) on data stored in a hash-based
data structure. In this application, a Prover holds the VRF secret
key and uses the VRF hashing to construct a hash-based data structure
on the input data. Due to the nature of the VRF, only the Prover can
answer queries about whether or not some data is stored in the data
structure. Anyone who knows the public VRF key can verify that the
Prover has answered the queries correctly. However no offline
inferences (i.e. inferences without querying the Prover) can be made
about the data stored in the data strucuture.
1.2. Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
1.3. Terminology
The following terminology is used through this document:
SK: The private key for the VRF.
PK: The public key for the VRF.
alpha: The input to be hashed by the VRF.
beta: The VRF hash output.
pi: The VRF proof.
Prover: The Prover holds the private VRF key SK and public VRF key
PK.
Verifier: The Verifier holds the public VRF key PK.
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2. VRF Algorithms
A VRF comes with a key generation algorithm that generates a public
VRF key PK and private VRF key SK.
A VRF hashes an input alpha using the private VRF key SK to obtain a
VRF hash output beta
beta = VRF_hash(SK, alpha)
The VRF_hash algorithm is deterministic, in the sense that it always
produces the same output beta given a pair of inputs (SK, alpha).
The private key SK is also used to construct a proof pi that beta is
the correct hash output
pi = VRF_prove(SK, alpha)
The VRFs defined in this document allow anyone to deterministically
obtain the VRF hash output beta directly from the proof value pi as
beta = VRF_proof2hash(pi)
Notice that this means that
VRF_hash(SK, alpha) = VRF_proof2hash(VRF_prove(SK, alpha))
The proof pi allows a Verifier holding the public key PK to verify
that beta is the correct VRF hash of input alpha under key PK. Thus,
the VRF also comes with an algorithm
VRF_verify(PK, alpha, pi)
that outputs VALID if beta=VRF_proof2hash(pi) is correct VRF hash of
alpha under key PK, and outputs INVALID otherwise.
3. VRF Security Properties
VRFs are designed to ensure the following security properties.
3.1. Full Uniqueness or Trusted Uniqueness
Uniqueness means that, for any fixed public VRF key and for any input
alpha, there is a unique VRF output beta that can be proved to be
valid. Uniqueness must hold even for an adversarial Prover that
knows the VRF secret key SK.
"Full uniqueness" states that a computationally-bounded adversary
cannot choose a VRF public key PK, a VRF input alpha, two different
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VRF hash outputs beta1 and beta2, and two proofs pi1 and pi2 such
that VRF_verify(PK, alpha, pi1) and VRF_verify(PK, alpha, pi2) both
output VALID.
A slightly weaker security property called "trusted uniquness"
sufficies for many applications. Trusted uniqueness is the same as
full uniqueness, but it must hold only if the VRF keys PK and SK were
generated in a trustworthy manner. In otherwords, uniqueness might
not hold if keys were generated in an invalid manner or with bad
randomness.
3.2. Full Collison Resistance or Trusted Collision Resistance
Like any cryprographic hash function, VRFs need to be collision
resistant. Collison resistance must hold even for an adversarial
Prover that knows the VRF secret key SK.
More percisely, "full collision resistance" states that it should be
computationally infeasible for an adversary to find two distinct VRF
inputs alpha1 and alpha2 that have the same VRF hash beta, even if
that adversary knows the secret VRF key SK.
For most applications, a slightly weaker security property called
"trusted collision resistance" suffices. Trusted collision
resistance is the same as collision resistance, but it holds only if
PK and SK were generated in a trustworthy manner.
3.3. Full Pseudorandomness or Selective Pseudorandomness
Pseudorandomness ensures that when an adversarial Verifier sees a VRF
hash output beta without its corresponding VRF proof pi, then beta is
indistinguishable from a random value.
More percisely, suppose the public and private VRF keys (PK, SK) were
generated in a trustworthy manner. Pseudorandomness ensures that the
VRF hash output beta (without its corresponding VRF proof pi) on any
adversarially-chosen "target" VRF input alpha looks indistinguishable
from random for any computationally bounded adversary who does not
know the private VRF key SK. This holds even if the adversary also
gets to choose other VRF inputs alpha' and observe their
corresponding VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the
"target" VRF input alpha at any time, even after it observes VRF
outputs beta' and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property which
suffices in many applications. Here, the adversary must choose the
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target VRF input alpha independently of the public VRF key PK, and
before it observes VRF outputs beta' and proofs pi' on inputs alpha'
of its choice.
It is important to remember that the VRF output beta does not look
random to the Prover, or to any other party that knows the private
VRF key SK! Such a party can easily distinguish beta from a random
value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that
knows valid VRF proof pi corresponding to the VRF input alpha, even
if this party does not know the private VRF key SK. Such a party can
easily distinguish beta from a random value by checking whether
VRF_verify(PK, alpha, pi) returns "VALID" and beta =
VRF_proof2hash(pi).
Also, the VRF output beta may not look random if VRF key generation
was not done in a trustworthy fashion. (For example, if VRF keys
were generated with bad randomness.)
3.4. An additional pseudorandomness property
[TODO: The following property is not needed for applications that use
VRFs to prevent enumeration of hash-based data structures. However,
we noticed that some other applications of VRF rely on this property.
As we have not yet found a formal definition of this property in the
literature, we write it down here. ]
Pseudorandomness, as defined in Section 3.3, does not hold if the VRF
keys were generated adversarially.
There is, however, a different type of pseudorandomness that could
hold even if the VRF keys are generated adversarially, as long as the
VRF input alpha is unpredictable. Suppose the VRF keys are generated
by an adversary. Then, a VRF hash output beta should look
pseudorandom to the adversary as long as (1) its corresponding VRF
hash alpha is chosen randomly and independently of the VRF key, (2)
alpha is unknown to the adversary, (3) the corresponding proof pi is
unknown to the adversary, and (4) the VRF public key chosen by the
adversary is valid.
[TODO: It should be possible to get the EC-VRF to satisfy this
property, as long as verifiers run an VRF_validate_key() key function
upon receipt of VRF public keys. However, we need to work out
exactly what properties are needed from the VRF public keys in order
for this property to hold. Some additional checks might need to be
added to the ECVRF_validate_key() function. Need to work out what
are these checks.]
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4. RSA Full Domain Hash VRF (RSA-FDH-VRF)
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies
the "trusted uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. Its security
follows from the standard RSA assumption in the random oracle model.
Formal security proofs are in [nsec5ecc].
The VRF computes the proof pi as a deterministic RSA signature on
input alpha using the RSA Full Domain Hash Algorithm [RFC8017]
parametrized with the selected hash algorithm. RSA signature
verification is used to verify the correctness of the proof. The VRF
hash output beta is simply obtained by hashing the proof pi with the
selected hash algorithm.
The key pair for RSA-FDH-VRF MUST be generated in a way that it
satisfies the conditions specified in Section 3 of [RFC8017].
In this document, the notation from [RFC8017] is used.
Parameters used:
(n, e) - RSA public key
K - RSA private key
k - length in octets of the RSA modulus n
Fixed options:
Hash - cryptographic hash function
hLen - output length in octets of hash function Hash
Constraints on options:
Cryptographic security of Hash is at least as high as the
cryptographic security level of the RSA key
Primitives used:
I2OSP - Coversion of a nonnegative integer to an octet string as
defined in Section 4.1 of [RFC8017]
OS2IP - Coversion of an octet string to a nonnegative integer as
defined in Section 4.2 of [RFC8017]
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RSASP1 - RSA signature primitive as defined in Section 5.2.1 of
[RFC8017]
RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of
[RFC8017]
MGF1 - Mask Generation Function based on a hash function as
defined in Section B.2.1 of [RFC8017]
4.1. RSA-FDH-VRF Proving
RSAFDHVRF_prove(K, alpha)
Input:
K - RSA private key
alpha - VRF hash input, an octet string
Output:
pi - proof, an octet string of length k
Steps:
1. EM = MGF1(alpha, k - 1)
2. m = OS2IP(EM)
3. s = RSASP1(K, m)
4. pi = I2OSP(s, k)
5. Output pi
4.2. RSA-FDH-VRF Proof To Hash
RSAFDHVRF_proof2hash(pi)
Input:
pi - proof, an octet string of length k
Output:
beta - VRF hash output, an octet string of length hLen
Steps:
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1. beta = Hash(pi)
2. Output beta
4.3. RSA-FDH-VRF Verifying
RSAFDHVRF_verify((n, e), alpha, pi)
Input:
(n, e) - RSA public key
alpha - VRF hash input, an octet string
pi - proof to be verified, an octet string of length n
Output:
"VALID" or "INVALID"
Steps:
1. s = OS2IP(pi)
2. m = RSAVP1((n, e), s)
3. EM = I2OSP(m, k - 1)
4. EM' = MGF1(alpha, k - 1)
5. If EM and EM' are equal, output "VALID"; else output "INVALID".
5. Elliptic Curve VRF (EC-VRF)
The Elliptic Curve Verifiable Random Function (EC-VRF) is a VRF that
satisfies the trusted uniqueness, trusted collision resistance, and
full pseudorandomness properties defined in Section 3. The security
of this VRF follows from the decisional Diffie-Hellman (DDH)
assumption in the random oracle model. Formal security proofs are in
[nsec5ecc].
Fixed options:
F - finite field
2n - length, in octets, of a field element in F
E - elliptic curve (EC) defined over F
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m - length, in octets, of an EC point encoded as an octet string
G - subgroup of E of large prime order
q - prime order of group G
cofactor - number of points on E divided by q
g - generator of group G
Hash - cryptographic hash function
hLen - output length in octets of Hash
Constraints on options:
Field elements in F have bit lengths divisible by 16
hLen is equal to 2n
Parameters used:
y = g^x - VRF public key, an EC point
x - VRF private key, an integer where 0 < x < q
Notation and primitives used:
p^k - when p is an EC point: point multiplication, i.e. k
repetitions of group operation on EC point p. when p is an
integer: exponentiation
|| - octet string concatenation
I2OSP - nonnegative integer conversion to octet string as defined
in Section 4.1 of [RFC8017]
OS2IP - Coversion of an octet string to a nonnegative integer as
defined in Section 4.2 of [RFC8017]
EC2OSP - conversion of EC point to an m-octet string as specified
in Section 5.5
OS2ECP - conversion of an m-octet string to EC point as specified
in Section 5.5. OS2ECP returns INVALID if the octet string does
not convert to a valid EC point.
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RS2ECP - conversion of a random 2n-octet string to an EC point as
specified in Section 5.5
5.1. EC-VRF Proving
Note: this function is made more efficient by taking in the public
VRF key y, as well as the private VRF key x.
ECVRF_prove(y, x, alpha)
Input:
y - public key, an EC point
x - private key, an integer
alpha - VRF input, an octet string
Output:
pi - VRF proof, octet string of length m+3n
Steps:
1. h = ECVRF_hash_to_curve(y, alpha)
2. gamma = h^x
3. choose a random integer nonce k from [0, q-1]
4. c = ECVRF_hash_points(g, h, y, gamma, g^k, h^k)
5. s = k - c*x mod q (where * denotes integer multiplication)
6. pi = EC2OSP(gamma) || I2OSP(c, n) || I2OSP(s, 2n)
7. Output pi
5.2. EC-VRF Proof To Hash
ECVRF_proof2hash(pi)
Input:
pi - VRF proof, octet string of length m+3n
Output:
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"INVALID", or
beta - VRF hash output, octet string of length 2n
Steps:
1. D = ECVRF_decode_proof(pi)
2. If D is "INVALID", output "INVALID" and stop
3. (gamma, c, s) = D
4. beta = Hash(EC2OSP(gamma^cofactor))
5. Output beta
5.3. EC-VRF Verifying
ECVRF_verify(y, pi, alpha)
Input:
y - public key, an EC point
pi - VRF proof, octet string of length 5n+1
alpha - VRF input, octet string
Output:
"VALID" or "INVALID"
Steps:
1. D = ECVRF_decode_proof(pi)
2. If D is "INVALID", output "INVALID" and stop
3. (gamma, c, s) = D
4. u = y^c * g^s (where * denotes EC point addition, i.e. a group
operation on two EC points)
5. h = ECVRF_hash_to_curve(y, alpha)
6. v = gamma^c * h^s (where * denotes EC point addition)
7. c' = ECVRF_hash_points(g, h, y, gamma, u, v)
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8. If c and c' are equal, output "VALID"; else output "INVALID"
[TODO: We could hash alpha instead of gamma. Doing so costs more
because alpha is longer, but may help salvage some security if
ECVRF_hash_to_curve is broken and not collision-resistant. Need to
analyze exact security preserved and decide if the tradeoff is worth
it.]
5.4. EC-VRF Auxiliary Functions
[TODO: analyze whether domain separation for hash functions used here
matters and if so how to ensure we have it.]
5.4.1. EC-VRF Hash To Curve
The ECVRF_hash_to_curve algorithm takes in an octet string alpha and
converts it to h, an EC point in G.
5.4.1.1. ECVRF_hash_to_curve1
The following ECVRF_hash_to_curve1(y, alpha) algorithm implements
ECVRF_hash_to_curve in a simple and generic way that works for any
elliptic curve.
The running time of this algorithm depends on alpha. For the
ciphersuites specified in Section 5.5, this algorithm is expected to
find a valid curve point after approximately two attempts (i.e., when
ctr=1) on average. See also [Icart09].
However, because the running time of algorithm depends on alpha, this
algorithm SHOULD be avoided in applications where it is important
that the VRF input alpha remain secret.
ECVRF_hash_to_curve1(y, alpha)
Input:
alpha - value to be hashed, an octet string
y - public key, an EC point
Output:
h - hashed value, a finite EC point in G
Steps:
1. ctr = 0
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2. pk = EC2OSP(y)
3. h = "INVALID"
4. While h is "INVALID" or h is EC point at infinity:
A. CTR = I2OSP(ctr, 4)
B. ctr = ctr + 1
C. attempted_hash = Hash(pk || alpha || CTR)
D. h = RS2ECP(attempted_hash)
E. If h is not "INVALID" and cofactor > 1, set h = h^cofactor
5. Output h
5.4.1.2. ECVRF_hash_to_curve2
For applications where VRF input alpha must be kept secret, the
following ECVRF_hash_to_curve algorithm MAY be used to used as
generic way to hash an octet string onto any elliptic curve.
[TODO: We should look into specifying the generic deterministic time
hash_to_curve algorithm from [Icart09]. We should also consider
Shallue-Woestijne-Ulas algorithm from [BCIMRT10] as well as Elligator
(for Ed25519) and Elligator Squared for more general curves. Some of
these options are summarized in an upcoming draft [SW18].]
5.4.2. EC-VRF Hash Points
ECVRF_hash_points(p_1, p_2, ..., p_j)
Input:
p_i - EC point in G
Output:
h - hash value, integer between 0 and 2^(8n)-1
Steps:
1. P = empty octet string
2. for p_i in [p_1, p_2, ... p_j]:
P = P || EC2OSP(p_i)
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3. h1 = Hash(P)
4. h2 = first n octets of h1
5. h = OS2IP(h2)
6. Output h
5.4.3. EC-VRF Decode Proof
ECVRF_decode_proof(pi)
Input:
pi - VRF proof, octet string (m+3n octets)
Output:
"INVALID", or
gamma - EC point
c - integer between 0 and 2^(8n)-1
s - integer between 0 and 2^(16n)-1
Steps:
1. let gamma', c', s' be pi split after m-th and m+n-th octet
2. gamma = OS2ECP(gamma')
3. if gamma = "INVALID" output "INVALID" and stop.
4. c = OS2IP(c')
5. s = OS2IP(s')
6. Output gamma, c, and s
5.5. EC-VRF Ciphersuites
This document defines EC-VRF-P256-SHA256 as follows:
o The EC group G is the NIST-P256 elliptic curve, with curve
parameters as specified in [FIPS-186-3] (Section D.1.2.3) and
[RFC5114] (Section 2.6). For this group, 2n = 32 and cofactor =
1.
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o The key pair generation primitive is specified in Section 3.2.1 of
[SECG1].
o EC2OSP is specified in Section 2.3.3 of [SECG1] with point
compression on. This implies m = 2n + 1 = 33.
o OS2ECP is specified in Section 2.3.4 of [SECG1].
o RS2ECP(h) = OS2ECP(0x02 || h). The input h is a 32-octet string
and the output is either an EC point or "INVALID".
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1.
This document defines EC-VRF-ED25519-SHA256 as follows:
o The EC group G is the Ed25519 elliptic curve with parameters
defined in Table 1 of [RFC8032]. For this group, 2n = 32 and
cofactor = 8.
o The key pair generation primitive is specified in Section 5.1.5 of
[RFC8032]
o EC2OSP is specified in Section 5.1.2 of [RFC8032]. This implies m
= 2n = 32.
o OS2ECP is specified in Section 5.1.3 of [RFC8032].
o RS2ECP is equivalent to OS2ECP.
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1.
[TODO: Should we add an EC-VRF-ED25519-SHA256-Elligator ciphersuite
where the Elligator hash function is used for ECVRF_hash-to-curve?
Should we add Elligator-Squared? We may want to consider options in
the upcoming draft [SW18].]
[TODO: Add an Ed448 ciphersuite? This is probably not needed...]
[NOTE: In the unlikely case that future versions of this spec use a
elliptic curve group G that does not also come with a specification
of the group generator g, then we can still have full uniqueness and
full collision resistance by adding an check to
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ECVRF_validate_key(PK) that ensures that g is a point on the elliptic
curve and g^cofactor is not the EC point at infinity.]
5.6. When the EC-VRF Keys are Untrusted
The EC-VRF as specified above is a VRF that satisfies the "trusted
uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. If the elliptic
curve parameters (including the generator g) are trusted, but the VRF
public key PK is not trusted, this VRF can be modified to
additionally satisfy "full uniqueness", and "full collision
resistance". This is done by additionally requiring the Verifier to
perform the following validation procedure upon receipt of the public
VRF key.
The Verifier MUST perform this validation procedure when the entity
that generated the public VRF key is untrusted. The public key MUST
NOT be used if this procedure returns "INVALID". Note well that this
procedure is not sufficient if the elliptic curve E or if g, the
generator of group G, is untrusted.
This procedure supposes that the public key provided to the Verifier
is an octet string. The procedure returns "INVALID" if the public
key in invalid. Otherwise, it returns y, the public key as an EC
point.
5.6.1. EC-VRF Validate Key
ECVRF_validate_key(PK)
Input:
PK - public key, an octet string
Output:
"INVALID", or
y - public key, an EC point
Steps:
1. y = OS2ECP(PK)
2. If y is "INVALID", output "INVALID" and stop
3. If y^cofactor is the EC point at infinty, output "INVALID" and
stop
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4. Output y
6. Implementation Status
An implementation of the RSA-FDH-VRF (SHA-256) and EC-VRF-P256-SHA256
was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5]
and is available at . The EC-
VRF implementation may be out of date as this spec has evolved.
The Key Transparency project at Google uses a VRF implemention that
is similar to the EC-VRF-P256-SHA256, with a few minor changes
including the use of SHA-512 instead of SHA-256. Its implementation
is available
An implementation by Yahoo! similar to the EC-VRF is available at
.
An implementation similar to EC-VRF is available as part of the
CONIKS implementation in Golang at .
Open Whisper Systems also uses a VRF very similar to EC-VRF-
ED25519-SHA512-Elligator, called VXEdDSA, and specified here:
7. Security Considerations
7.1. Key Generation
Applications that use the VRFs defined in this document MUST ensure
that that the VRF key is generated correctly, using good randomness.
7.1.1. Uniqueness and collision resistance with untrusted keys
The EC-VRF as specified in Section 5.1-Section 5.5 statisfies the
"trusted uniqueness" and "trusted collision resistance" properties as
long as the VRF keys are generated correctly, with good randomness.
If the Verifier trusts the VRF keys are generated correctly, it MAY
use the public key y as is.
However, if the EC-VRF uses keys that could be generated
adversarially, then the the Verfier MUST first perform the validation
procedure ECVRF_validate_key(PK) (specified in Section 5.6) upon
receipt of the public key PK as an octet string. If the validation
procedure outputs "INVALID", then the public key MUST not be used.
Otherwise, the procedure will output a valid public key y, and the
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EC-VRF with public key y satisfies the "full uniqueness" and "full
collision resistance" properties.
The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted
collision resistance" properties as long as the VRF keys are
generated correctly, with good randomness. These properties may not
hold if the keys are generated adversarially (e.g., if RSA is not
permutation). Meanwhile, the "full uniqueness" and "full collision
resistance" are properties that hold even if VRF keys are generated
by an adversary. The RSA-FDH-VRF defined in this document does not
have these properties. However, if adversarial key generation is a
concern, the RSA-FDH-VRF may be modifed to have these properties by
adding additional cryptographic checks that its public key has the
right form. These modifications are left for future specification.
7.1.2. Pseudorandomness with untrusted keys
Without good randomness, the "pseudorandomness" properties of the VRF
may not hold. Note that it is not possible to guarantee
pseudorandomness in the face of adversarially generated VRF keys.
This is because an adversary can always use bad randomness to
generate the VRF keys, and thus, the VRF output may not be
pseudorandom.
7.2. Selective vs Full Pseudorandomness
[nsec5ecc] presents cryptographic reductions to an underlying hard
problem (e.g. Decisional Diffie Hellman for the EC-VRF, or the
standard RSA assumption for RSA-FDH-VRF) that prove the VRFs
specificied in this document possess full pseudorandomness as well as
selective pseudorandomness. However, the cryptographic reductions
are tighter for selective pseudorandomness than for full
pseudorandomness. This means the the VRFs have quantitavely stronger
security guarentees for selective pseudorandomness.
Applications that are concerned about tightness of cryptographic
reductions therefore have two options.
o They may choose to ensure that selective pseudorandomness is
sufficient for the application. That is, that pseudorandomness of
outputs matters only for inputs that are chosen independently of
the VRF key.
o If full pseudorandomness is required for the application, the
application may increase security parameters to make up for the
loose security reduction. For RSA-FDH-VRF, this means increasing
the RSA key length. For EC-VRF, this means increasing the
cryptographic strength of the EC group G. For both RSA-FDH-VRF
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and EC-VRF the cryptographic strength of the hash function Hash
may also potentially need to be increased.
7.3. Proper randomness for EC-VRF
Applications that use the EC-VRF defined in this document MUST ensure
that the random nonce k used in the ECVRF_prove algorithm is chosen
with proper randomness. Otherwise, an adversary may be able to
recover the private VRF key x (and thus break pseudorandomness of the
VRF) after observing several valid VRF proofs pi.
[TODO: In the next version of the draft we should add a specification
for nonce generation.]
7.4. Timing attacks
The EC-VRF_hash_to_curve algorithm defined in Section 5.4.1.1 SHOULD
NOT be used in applications where the VRF input alpha is secret and
is hashed by the VRF on-the-fly. This is because the EC-
VRF_hash_to_curve algorithm's running time depends on the VRF input
alpha, and thus creates a timing channel that can be used to learn
information about alpha. That said, for most inputs the amount of
information obtained from such a timing attack is likely to be small
(1 bit, on average), since the algorithm is expected to find a valid
curve point after only two attempts. However, there might be inputs
which cause the algorithm to make many attempts before it finds a
valid curve point; for such inputs, the information leaked in a
timing attack will be more than 1 bit.
8. Change Log
Note to RFC Editor: if this document does not obsolete an existing
RFC, please remove this appendix before publication as an RFC.
00 - Forked this document from draft-goldbe-vrf-01.
01 - Minor updates, mostly highlighting TODO items.
9. Contributors
This document also would not be possible without the work of Moni
Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf
Ziv (Facebook). Shumon Huque (Salesforce) and David C. Lawerence
(Akamai) provided valuable input to this draft.
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10. References
10.1. Normative References
[FIPS-186-3]
National Institute for Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-3, June 2009.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC5114] Lepinski, M. and S. Kent, "Additional Diffie-Hellman
Groups for Use with IETF Standards", RFC 5114,
DOI 10.17487/RFC5114, January 2008,
.
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
.
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
.
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
[SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", Version 2.0, May 2009,
.
10.2. Informative References
[BCIMRT10]
Brier, E., Coron, J., Icart, T., Madore, D., Randriam, H.,
and M. Tibouchi, "Efficient Indifferentiable Hashing into
Ordinary Elliptic Curves", in CRYPTO, 2010.
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[I-D.vcelak-nsec5]
Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
Existence", draft-vcelak-nsec5-06 (work in progress),
January 2018.
[Icart09] Icart, T., "How to Hash into Elliptic Curves", in CRYPTO,
2009.
[MRV99] Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random
Functions", in FOCS, 1999.
[nsec5ecc]
Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J.,
Naor, M., Reyzin, L., and S. Goldberg, "Making NSEC5
Practical for DNSSEC", in ePrint Cryptology Archive
2017/099, February 2017,
.
[SW18] Sullivan, E. and C. Wood, "Hashing to Elliptic Curves",
2018.
Authors' Addresses
Sharon Goldberg
Boston University
111 Cummington St, MCS135
Boston, MA 02215
USA
EMail: goldbe@cs.bu.edu
Leonid Reyzin
Boston University
111 Cummington St, MCS135
Boston, MA 02215
USA
EMail: reyzin@bu.edu
Dimitrios Papadopoulos
Hong Kong University of Science and Techology
Clearwater Bay
Hong Kong
EMail: dipapado@cse.ust.hkbu.edu
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Jan Vcelak
NS1
16 Beaver St
New York, NY 10004
USA
EMail: jvcelak@ns1.com
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